Count outcomes without listing them. Build from the multiplication principle to sampling with and without replacement and the binomial coefficient, then decide which count a problem needs. Finish with arrangements of repeated objects and the binomial theorem.
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Lesson 01 · The multiplication principle
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5 lessons you can take today, and 3 more being written. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.
01
The multiplication principle
8 questions
Learn how to count the outcomes of a multi-stage choice by multiplying the number of options at each stage, and when that applies.
Covers
Counting sequential choices
When multiplication applies
02
Sampling with replacement
12 questions
Learn how to count outcomes when objects look alike by labelling them, count ordered samples drawn with replacement, and count the subsets of a set.
Covers
Labelling objects
Sampling with replacement
Counting subsets
03
Sampling without replacement
12 questions
Learn how to count ordered samples drawn without replacement, count the arrangements of a set of objects, and decide whether a scenario is sampled with or without replacement.
Covers
Sampling without replacement
Permutations
With or without?
04
The binomial coefficient
12 questions
Learn how to count selections where order does not matter by counting in order and correcting for the overcount, and how to name and evaluate that count as the binomial coefficient.
Covers
Adjusting for overcounting
Counting groups of
The binomial coefficient
05
Choosing a counting model
6 questions
Learn how to choose the right way to count a selection of objects by asking whether an object can be drawn more than once and whether the outcome records the order of the draws.
Covers
The sampling table
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Properties of the binomial coefficient
In drafting
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Arrangements with repeated objects
In drafting
—
The binomial theorem
In drafting
This unit is part of our Essential Probability & Statistics for ML learning path, which contains 23 lessons. Every one of them is drawn below.
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